Let P_n be the space of functions defined on [1, 1] that can be described by polynomials of degree less of equal to n with coefficients in R. P_n is a linear space in the sense of linear algebra, in particular, for p, q P_n and a R, also p + q and ap are in P_n. Since the monomials {1, x, x² , . . . , xⁿ} are a basis for P_n, the dimension of that space is n + 1.

(a) Show that for pairwise distinct points To, T1, . . ., In E [-1, 1], the Lagrange polyno- mials Lk(x) are in Pn, and that they are linearly independent, that is, for a linear combination of the zero polynomial with Lagrange polynomials with coefficients ck, i.e., _ ax LA(x) = 0 (the zero polynomial) *=0 necessarily follows that an = 01 = ... = (n = 0. Note that this implies that the (n + 1) Lagrange polynomials also form a basis of Pn. (b) Since both the monomials and the Lagrange polynomials are a basis of Pn, each p E Pn can be written as linear combination of monomials as well as Lagrange polynomials. i.e., p(x) =DaxLK(x) =Biz*, *=0 (1) *=0 with appropriate coefficients ak, Bk E R. As you know from basic matrix theory, there exists a basis transformation matrix that converts the coefficients o = (On, . . . , On)" to the coefficients B = (Bo, . ..,Bn)T. Show that this basis transformation matrix is given by the so-called Vandermonde matrix V E R"+1xn+1 given by TO V = I1 In . . . i.e., the relation between o and / in (1) is given by o = VB. An easy way to see this is to choose appropriate z in (1).{c} Note that since V transforms one basis into another basis+ it must be an invertible matrix. Let us compute the condition number of V numerically.1 Compute the 2 based condition number 52(1'} for n = 5,10,20,30 with uniforme spaced nodes 1."; = -1 + {it'llfm 1'. = l],...,n.. Based on the condition numbers. can this basis transformation be performed accurately